Unique Continuation And Inverse Source Problem For Helmholtz Equation

Scattering problems have been well-investigated in mathematical physics. Their backgrounds are propagation, scattering, reflection and diffraction of acoustic and electromagnetic waves. The wave equation is introduced to describe the wave phe-nomenon and Helmholtz equation focuses on the time-harmonic wave. One of the key topics in scattering problems relies on knowledge of the Helmholtz equation. In this dissertation we concentrate on the unique continuation and inverse source problems of Helmholtz equations. By utilizing analytic property of solutions of the Helmholtz exterior problem, we prove the unique continuation on an analytic curve for this solution and provide a Holder type conditional stability estimate. The rest of the dissertation then focuses on the inverse source problem of Helmholtz equa-tions. Because of the non-radiating source phenomena, boundary observation of single or finite number frequencies (wavenumbers) can not determine the source term uniquely. By the unique continuation of Helmholtz equations, we prove that partial multi-frequency observation data on a closed analytic curve can uniquely identify the interior source. Later on we propose two recursive algorithm aiming at the recovery of full Fourier coefficient of the source function. Convergence and error estimate of these algorithms are provide. Numerical examples show that the recursive algorithms will retrieve local information from observation data of different frequencies and build up a good approximation within multi-frequency setting.Chapter 1 denotes to the brief introduction of ill-posed problems and back-ground of Helmholtz equations.In Chapter 2, we provide some basic knowledge which will be extensively re-called in current dissertation, including classic regularization schemes, far field pat-tern of Helmholtz equations, spherical harmonics and Bessel functions, basic poten-tial theory and well-posedness of the Helmholtz exterior problem.In Chapter 3 we discuss the unique continuation of Helmholtz equations. By considering the analytic extension of the holomorphic functions and finite open covering theorem, we prove the unique continuation theory and obtain a Holder type conditional stability estimate. Based on the collocation method, we propose Tikhonov regularization to realize the unique continuation numerically.In Chapter 4, we extensively discuss the inverse source problem of Helmholtz equations and its solvability. Such a problem is difficult in inverse scattering theory. Because of the non-radiating source phenomena, one can not uniquely identify the source function by boundary observation of single or finite number frequencies. By fixing the observation and source domains, we build up Fourier series as a set of basis functions of L2 space. These Fourier series is generated by solving the Dirich-let eigenvalue problem of Laplace equations and appears in the form of spherical harmonics and Bessel functions. We prove the uniqueness of inverse source prob-lems with knowledge of all the boundary observation for Helmholtz equations coping with all the wavenumbers coinciding with Dirichlet eigenvalues. Finally we build up two recursive algorithms aiming at the recovery of full Fourier coefficients of the source function. Convergence and error estimate of these algorithms are provide. Numerical examples verify the robustness of our proposed algorithms.